Dynamical systems around the Rauzy gasket and their ergodic properties

TL;DR

This paper establishes a connection between two different theories of measured foliations and interval exchange transformations, showing they describe the same objects and providing a translation between their frameworks.

Contribution

It proves that the French and Russian constructions of measured foliations are equivalent and offers an explicit dictionary linking the two approaches.

Findings

Both theories describe the same objects in the simplest case.

The space of minimal parameters is the same, called the Rauzy gasket.

The paper provides an explicit translation between the two frameworks.

Abstract

At the beggining of the 80's, H.Masur and W.Veech started the study of generic properties of interval exchange transformations proving that almost every such transformation is uniquely ergodic. About the same time, S.Novikov's school and French mathematicians independently discovered very intriguing phenomena for classes of measured foliations on surfaces and respective IETs. For instance, minimality is exceptional in these families. A precise version of this statement is a conjecture by Novikov. The French and Russian constructions are very different ones. Nevertheless, in the most simple situation (surfaces of genus three with two singularities) it was recently observed that both foliations share the same type of properties. For instance, the space of minimal parameters is the same, called the Rauzy gasket. However, the precise connection between these two series of works was rather unclear. The aim of this paper is to prove that both theories describe in different languages the same objects. This text provides an explicit dictionary between both constructions.